Prepare a medium-hot fire in a charcoal grill with a cover, or preheat a gas grill to medium-high.

To make the grilled jicama and onion, peel and slice the jicama and onion into 1/8-inch-thick slices. Place in a bowl and add the mirin and soy sauce. Rotate around so that all sides of the slices are moistened.

To make the patties, combine the beef, ginger, White Zinfandel, soy sauce, mirin, onion salt, garlic salt and pepper in a large bowl. Handling the meat as little as possible to avoid compacting it, mix well. Divide the mixture into 4 equal portions and form the portions into patties to fit the rolls.

When the grill is ready, brush the grill rack with vegetable oil. Place the patties on the rack, cover, and cook, turning once, until done to preference, 5 to 7 minutes on each side for medium. Grill the jicama and onion slices until lightly browned, about 5 minutes on each side. During the last few minutes of cooking, place the rolls, cut side down, on the outer edges of the rack to toast lightly.

To assemble the burgers, spread the butter over the cut sides of the rolls. On each roll bottom, place a jicama slice, a patty, and an onion slice. Add the roll tops and serve.